A188181 -id:A188181 - cp's OEIS frontend

A002838 Balancing weights on the integer line.

1, 2, 5, 12, 32, 94, 289, 910, 2934, 9686, 32540, 110780, 381676, 1328980, 4669367, 16535154, 58965214, 211591218, 763535450, 2769176514, 10089240974, 36912710568, 135565151486, 499619269774, 1847267563742, 6850369296298

Offset: 1

N. J. A. Sloane

Also number of partitions of n(n+1)/2 into up to n parts each no greater than n+1, partitions of n(n+3)/2 into exactly n parts each no greater than n+2 and partitions of n(n+1) into exactly n distinct parts each no greater than 2n+1, thus providing balancing solutions for n weights in distinct integer positions on [ -n,n] with a pivot at 0. - Henry Bottomley, Aug 09 2002

Is this a shifted version of A076822? - Vladimir Reshetnikov, Oct 06 2016

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. E. Odeh and E. J. Cockayne, Balancing weights on the integer line, J. Combin. Theory, 7 (1969), 130-135.

Cf. A047997, A076822, A188181 (columns 1, 2).

(* This program is not convenient for large values of n *) a[n_] := Length[ IntegerPartitions[n*(n+1)/2, n, Range[n+1]]]; Table[ Print[{n, an = a[n]}]; an, {n, 1, 16}] (* Jean-François Alcover, Jan 02 2013 *)

a(n) = A047997(n, n) = A067059(n, n+1). a(n) tends towards (sqrt(12)/Pi)*4^n/n^2 and something like (sqrt(12)/Pi)*4^n/(n^2+1.85*n+0.8) seems to give an even closer approximation. - Henry Bottomley, Aug 09 2002

More terms from Henry Bottomley, Aug 09 2002

A076822 Number of partitions of the n-th triangular number involving only the numbers 1..n and with exactly n terms.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 32, 94, 289, 910, 2934, 9686, 32540, 110780, 381676, 1328980, 4669367, 16535154, 58965214, 211591218, 763535450, 2769176514, 10089240974, 36912710568, 135565151486, 499619269774, 1847267563742, 6850369296298

Offset: 0

Jon Perry, Nov 19 2002

nonn

Asymptotic to (sqrt(3)/(2*Pi))*(4^n/n^2). It is the number of lattice paths from (0,0) to (n,n-1) with steps only to the right or upward and having area n(n-1)/2 between the path and the x-axis. In the reference by Takács use formula (77) with a=n, b=n(n-1)/2 and then Stirling's formula. - Kent E. Morrison, May 28 2016

a(n) is the number of fair dice with n sides and expected value (n+1)/2 with distinct composition of numbers between 1 and n. - Felix Huber, Aug 02 2024

			a(4)=5 as T(4)=10= 1+1+4+4 =1+2+3+4 = 1+3+3+3 = 2+2+2+4 = 2+2+3+3.

Max Alekseyev and Alois P. Heinz, Table of n, a(n) for n = 0..240 (terms n=1..100 from Max Alekseyev)
L. Takács, Some asymptotic formulas for lattice paths, J. Statist. Plann. Inference, 14 (1986), 123-142.

Cf. A067059, A047993, A039744.
Cf. A002838. [From R. J. Mathar, Sep 20 2008]
Cf. A188181 (columns 1, 2).

ccc=new Array(); cccc=0;
for (n=1; n<11; n++)
{
    str='cc=0; for (i1=1; i1<'+(n+1)+'; i1++)';
    str2='i1';
    str3='i1';
    tn=1;
    for (i=2; i<=n; i++)
    {
        str+='for (i'+i+'=i'+(i-1)+'; i'+i+'<'+(n+1)+'; i'+i+'++)';
        str2+='+i'+i;
        str3+=', ", ", i'+i;
        tn+=i;
    }
    str+='if ('+str2+'=='+tn+') document.print(++cc, ":", '+str3+', "
")';
    eval(str);
    ccc[cccc++ ]=cc;
    document.print('****
');
}
document.write(ccc);

f[n_] := Block[{p = IntegerPartitions[n(n + 1)/2, n]}, Length[ Select[p, Length[ # ] == n &]]]; Table[ f[n], {n, 1, 13}]

a(n) = A067059(n,n+1); also a(n) = T[n*(n-1)/2, n-1, n] with T[ ] defined as in A047993. - Martin Fuller, Jun 27 2006

Edited and extended to 12 terms by Robert G. Wilson v, Nov 23 2002
Further terms from Max Alekseyev, May 24 2007
a(0)=1 prepended by Alois P. Heinz, May 28 2016

A188183 Number of strictly increasing arrangements of 5 numbers in -(n+3)..(n+3) with sum zero.

Original entry on oeis.org

12, 32, 73, 141, 252, 414, 649, 967, 1394, 1944, 2649, 3523, 4604, 5910, 7483, 9343, 11538, 14090, 17053, 20451, 24342, 28754, 33751, 39361, 45654, 52662, 60459, 69079, 78602, 89064, 100551, 113101, 126804, 141702, 157891, 175413, 194370, 214808

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Row 5 of A188181

			Some solutions for n=5
.-5...-8...-7...-8...-6...-4...-8...-6...-8...-5...-8...-7...-6...-6...-8...-7
.-3...-3...-4...-6...-2...-3...-7...-5...-2...-3...-2...-4...-5...-5...-1...-6
.-1...-2....0....0...-1...-1....4...-2....2...-1....1...-2....2...-3....1....1
..3....5....4....6....2....0....5....6....3....1....3....6....3....6....3....4
..6....8....7....8....7....8....6....7....5....8....6....7....6....8....5....8

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n)=2*a(n-1)-a(n-3)-2*a(n-5)+2*a(n-6)+a(n-8)-2*a(n-10)+a(n-11).

Empirical: a(n) = 427*n^2/144 +155*n/32 +5501/1728+23*n^4/288 +115*n^3/144 -3*(-1)^n*n/32-15*(-1)^n/64 +A057077(n+1)/8 -2*A061347(n+1)/27; g.f. -x*(12 +8*x +9*x^2 +7*x^3 +2*x^4 +7*x^5 +2*x^6 +3*x^7 -2*x^8 -5*x^9 +3*x^10) / ( (x^2+1) *(1+x+x^2) *(1+x)^2 *(x-1)^5 ). - R. J. Mathar, Mar 26 2011

A188185 Number of strictly increasing arrangements of 7 numbers in -(n+5)..(n+5) with sum zero.

Original entry on oeis.org

94, 289, 734, 1656, 3370, 6375, 11322, 19138, 30982, 48417, 73316, 108108, 155646, 219489, 303748, 413442, 554256, 733005, 957332, 1236222, 1579666, 1999265, 2507780, 3119876, 3851588, 4721127, 5748298, 6955424, 8366614, 10008857

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Row 7 of A188181

			Some solutions for n=5
.-9...-7...-9..-10..-10..-10...-7...-9..-10..-10..-10..-10..-10...-9...-7..-10
.-7...-4...-7...-6...-7...-4...-6...-7...-3...-8...-5...-8...-8...-6...-6...-6
.-3...-2...-2...-5...-3...-3...-4...-4...-1...-2...-1...-7...-5...-1...-3...-5
.-1...-1....0....2....2...-1...-3....1....0...-1....1....3...-1....1...-2....1
..5....0....3....4....3....3....4....2....1....5....2....6....6....2....0....4
..6....6....6....7....6....7....7....8....4....7....4....7....8....4....8....7
..9....8....9....8....9....8....9....9....9....9....9....9...10....9...10....9

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n)=2*a(n-1)-a(n-3)-a(n-5)+a(n-6)-2*a(n-7)+2*a(n-8)+a(n-9)-a(n-13)-2*a(n-14)+2*a(n-15)-a(n-16)+a(n-17)+a(n-19)-2*a(n-21)+a(n-22).

Empirical: G.f. -x*(94 +101*x +156*x^2 +282*x^3 +347*x^4 +463*x^5 +423*x^6 +497*x^7 +393*x^8 +285*x^9 +180*x^10 +99*x^11 +17*x^12 -25*x^13 +47*x^14 -19*x^15 +25*x^16 +4*x^17 +22*x^18 -7*x^19 -44*x^20 +24*x^21) / ( (x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^2+1) *(1+x+x^2)^2 *(1+x)^3 *(x-1)^7 ). - R. J. Mathar, Mar 26 2011

A188174 Number of strictly increasing arrangements of n numbers in -(2n-2)..(2n-2) with sum zero.

Original entry on oeis.org

1, 2, 8, 43, 252, 1636, 11322, 81805, 610358, 4672836, 36525052, 290399543, 2342043630, 19118822650, 157709971144, 1312796873451, 11015236989702, 93078467831012, 791462018168032, 6767910089071288, 58167781282121262

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Diagonal of A188181

			Some solutions for n=5
.-8...-7...-4...-6...-6...-7...-8...-8...-5...-5...-7...-6...-5...-7...-4...-7
.-6...-4...-2...-5...-5...-2...-1...-3...-2...-2...-5...-4...-3...-3...-3...-4
.-1....1....0...-3....1....1....0...-1...-1...-1....0...-1...-1....2...-1....1
..7....4....2....6....3....3....1....5....0....1....4....3....3....3....3....2
..8....6....4....8....7....5....8....7....8....7....8....8....6....5....5....8

R. H. Hardin, Table of n, a(n) for n = 1..32

A188175 Number of strictly increasing arrangements of n numbers in -(n+1)..(n+1) with sum zero.

Original entry on oeis.org

1, 3, 8, 24, 73, 227, 734, 2430, 8150, 27718, 95514, 332578, 1168261, 4136477, 14749992, 52925886, 190973410, 692583902, 2523265494, 9231352260, 33901898722, 124940568222, 461938289518, 1713007181342, 6369928427268, 23747917426918

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Column 3 of A188181.

Apparently a(n) = A046919(n+3) for 1 <= n <= 30. - Georg Fischer, Oct 24 2018

			Some solutions for n=5
.-6...-5...-3...-4...-6...-4...-5...-4...-5...-3...-5...-6...-6...-6...-6...-5
.-2...-1...-2...-3...-3...-2...-4...-3...-3...-2...-3...-1...-1...-3...-2...-3
..0....0...-1...-1....0....0....0....0....0...-1....0....0....0....1....1....1
..2....1....2....3....3....2....3....1....3....1....2....3....2....3....3....2
..6....5....4....5....6....4....6....6....5....5....6....4....5....5....4....5

R. H. Hardin, Table of n, a(n) for n = 1..30

Cf. A046919.

A188176 Number of strictly increasing arrangements of n numbers in -(n+2)..(n+2) with sum zero.

Original entry on oeis.org

1, 4, 13, 43, 141, 480, 1656, 5744, 20094, 70922, 252117, 901723, 3243531, 11728606, 42611990, 155484150, 569585274, 2094177794, 7725489976, 28588154238, 106095329440, 394792407478, 1472734812454, 5506709078310, 20635115495666

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Column 4 of A188181

			Some solutions for n=5
.-4...-5...-7...-5...-7...-6...-5...-3...-6...-6...-6...-4...-7...-7...-5...-6
.-3...-1...-2...-3...-5...-5...-4...-2...-4...-2...-3...-3...-2...-4...-4...-4
.-2....0....1...-2...-1....1...-1....0...-1....0...-1...-1....0....0...-3....2
..2....1....2....4....6....3....3....2....5....3....3....3....3....4....5....3
..7....5....6....6....7....7....7....3....6....5....7....5....6....7....7....5

R. H. Hardin, Table of n, a(n) for n = 1..29

A188177 Number of strictly increasing arrangements of n numbers in -(n+3)..(n+3) with sum zero.

Original entry on oeis.org

1, 5, 18, 69, 252, 920, 3370, 12346, 45207, 165821, 609734, 2247151, 8300708, 30734578, 114067100, 424306230, 1581788460, 5909243454, 22120325320, 82963636468, 311732564498, 1173377033602, 4424032482238, 16706682557212

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Column 5 of A188181

			Some solutions for n=6
.-8...-8...-9...-8...-9...-8...-9...-6...-9...-8...-4...-9...-4...-9...-8...-8
.-4...-7...-1...-3...-2...-7...-3...-2...-3...-7...-3...-8...-3...-6...-5...-3
.-3...-1....0...-2....0...-4...-2...-1...-1...-3...-2....0...-2....1...-3...-1
.-1....4....1....0....1....2....2....0....3....2....2....3...-1....3....0....0
..7....5....3....5....2....8....5....1....4....7....3....6....3....5....7....5
..9....7....6....8....8....9....7....8....6....9....4....8....7....6....9....7

R. H. Hardin, Table of n, a(n) for n = 1..28

A188178 Number of strictly increasing arrangements of n numbers in -(n+4)..(n+4) with sum zero.

Original entry on oeis.org

1, 6, 25, 104, 414, 1636, 6375, 24591, 94257, 360002, 1371535, 5216252, 19819846, 75277670, 285893124, 1085984406, 4126777398, 15690144942, 59691687518, 227248928264, 865788366806, 3301087870766, 12596381911969, 48104063936789

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Column 6 of A188181

			Some solutions for n=5
.-7...-4...-9...-9...-8...-8...-6...-6...-7...-5...-7...-9...-5...-9...-9...-5
.-5...-2...-2...-6...-4...-6...-4...-5...-6...-3...-4...-1...-3...-5...-5...-4
..0....0....2....3....1...-3...-2....0....0...-1...-3....0....0....2...-2....0
..5....1....4....5....2....8....4....3....5....4....6....3....3....4....7....4
..7....5....5....7....9....9....8....8....8....5....8....7....5....8....9....5

R. H. Hardin, Table of n, a(n) for n = 1..27

A188179 Number of strictly increasing arrangements of n numbers in -(n+5)..(n+5) with sum zero.

Original entry on oeis.org

1, 7, 32, 150, 649, 2739, 11322, 46029, 184717, 734517, 2900900, 11396054, 44585180, 173885716, 676537634, 2627337510, 10188889502, 39470938574, 152787805216, 591094837268, 2285918145819, 8838165075791, 34167434076214, 132084761317107

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Column 7 of A188181

			Some solutions for n=5
.-4...-9...-9...-8...-7...-7..-10...-7..-10..-10...-9...-9..-10...-9...-6...-7
.-3...-7...-5...-5...-3...-3...-5...-3....0...-8...-4...-3...-7...-4...-2...-2
.-1....0...-2...-3....2...-1....2....1....1....2....1....0....3...-2....0....0
..0....7....7....7....3....3....4....3....4....6....5....2....4....5....3....2
..8....9....9....9....5....8....9....6....5...10....7...10...10...10....5....7

R. H. Hardin, Table of n, a(n) for n = 1..26

cp's OEIS Frontend

A002838 Balancing weights on the integer line.

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A076822 Number of partitions of the n-th triangular number involving only the numbers 1..n and with exactly n terms.

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A188183 Number of strictly increasing arrangements of 5 numbers in -(n+3)..(n+3) with sum zero.

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A188185 Number of strictly increasing arrangements of 7 numbers in -(n+5)..(n+5) with sum zero.

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A188174 Number of strictly increasing arrangements of n numbers in -(2n-2)..(2n-2) with sum zero.

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A188175 Number of strictly increasing arrangements of n numbers in -(n+1)..(n+1) with sum zero.

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A188176 Number of strictly increasing arrangements of n numbers in -(n+2)..(n+2) with sum zero.

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A188177 Number of strictly increasing arrangements of n numbers in -(n+3)..(n+3) with sum zero.

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A188178 Number of strictly increasing arrangements of n numbers in -(n+4)..(n+4) with sum zero.

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A188179 Number of strictly increasing arrangements of n numbers in -(n+5)..(n+5) with sum zero.

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